LPI depends on how dense the wood is.  Eng. walnut will handle up to 26 LPI if it is from Ca. rather than England. American black walnut will generally speaking will also handle 22 to 24 LPI IF it is from an area where there isn't much rain fall. Softer walnuts such as claro usually doesn't accept fine LPI as it tends to fuzz due to its soft nature.    FWIW,     Ken

Listen to Doug ----- he's a wizz at that checkering stuff. Better yet, let him do it if you want a pro. job.   Ken

I personally prefer to space my checkering @ 22 LPI, and then deepen with a carbide Pointer-Long cutter mounted in a see-thru adjustable Brownell handle. With my somewhat poor spacing technique, the 22-LPI tool actually ends up cutting about 23 LPI and sometimes more. I also use a veiner in the tight places, and I make sure each line is STRAIGHT before spacing the next. A 90-degree bent riffler in the fine cut is a good tool for the final polish, and can be a lifesaver with fuzzy wood. Also many checkerers are now using Super Glue to stabilize problem woods as well as to strengthen the diamonds. Simply flow the liquid into the checkering and brush out the excess with a toothbrush or similar, being careful to not let the glue attack the bristles, then go back over it with the tools as necessary after drying.

I strongly suggest that you start with about 20 LPI and practice on a finished piece such as an old stock. An old Mauser stock is best IMO since they were made from juglans regia and will respond well to the tools, cutting very cleanly and so allowing you to concentrate upon your technique instead of trying to fight a softer wood. And BTW 20 LPI is about what most of the older rifles had, at least most of the ones I've owned. Also most of the old checkering seems to have the 3:1 length-to-width diamond ratio, but for a somewhat more pleasing look I prefer the 3.5:1 ratio.

Checkering is a subject unto itself and IMO you can't do better than to consult Monte Kennedy's great book Checkering and Carving of Gunstocks. I particularly like the various Dale Goens patterns and have used them extensively in my own stockwork.
Good luck, Joe